↑

Properties

Label 2-2e4-16.13-c13-0-22
Degree $2$
Conductor $16$
Sign $-0.448 + 0.893i$
Analytic cond. $17.1569$
Root an. cond. $4.14209$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (89.0 − 16.0i)2-s + (−217. − 217. i)3-s + (7.67e3 − 2.86e3i)4-s + (−3.64e4 + 3.64e4i)5-s + (−2.28e4 − 1.58e4i)6-s − 3.97e5i·7-s + (6.37e5 − 3.78e5i)8-s − 1.49e6i·9-s + (−2.65e6 + 3.82e6i)10-s + (2.25e6 − 2.25e6i)11-s + (−2.29e6 − 1.04e6i)12-s + (−1.53e7 − 1.53e7i)13-s + (−6.38e6 − 3.53e7i)14-s + 1.58e7·15-s + (5.07e7 − 4.39e7i)16-s − 1.30e8·17-s + ⋯
L(s)  = 1  + (0.984 − 0.177i)2-s + (−0.172 − 0.172i)3-s + (0.937 − 0.349i)4-s + (−1.04 + 1.04i)5-s + (−0.200 − 0.139i)6-s − 1.27i·7-s + (0.860 − 0.509i)8-s − 0.940i·9-s + (−0.841 + 1.21i)10-s + (0.384 − 0.384i)11-s + (−0.221 − 0.101i)12-s + (−0.882 − 0.882i)13-s + (−0.226 − 1.25i)14-s + 0.359·15-s + (0.756 − 0.654i)16-s − 1.31·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.448 + 0.893i$
Analytic conductor: \(17.1569\)
Root analytic conductor: \(4.14209\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :13/2),\ -0.448 + 0.893i)\)

Particular Values

\(L(7)\) \(\approx\) \(1.11075 - 1.79949i\)
\(L(\frac12)\) \(\approx\) \(1.11075 - 1.79949i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-89.0 + 16.0i)T \)
good3 \( 1 + (217. + 217. i)T + 1.59e6iT^{2} \)
5 \( 1 + (3.64e4 - 3.64e4i)T - 1.22e9iT^{2} \)
7 \( 1 + 3.97e5iT - 9.68e10T^{2} \)
11 \( 1 + (-2.25e6 + 2.25e6i)T - 3.45e13iT^{2} \)
13 \( 1 + (1.53e7 + 1.53e7i)T + 3.02e14iT^{2} \)
17 \( 1 + 1.30e8T + 9.90e15T^{2} \)
19 \( 1 + (-1.52e8 - 1.52e8i)T + 4.20e16iT^{2} \)
23 \( 1 + 1.11e9iT - 5.04e17T^{2} \)
29 \( 1 + (-2.05e9 - 2.05e9i)T + 1.02e19iT^{2} \)
31 \( 1 + 4.46e9T + 2.44e19T^{2} \)
37 \( 1 + (-1.19e10 + 1.19e10i)T - 2.43e20iT^{2} \)
41 \( 1 - 4.82e10iT - 9.25e20T^{2} \)
43 \( 1 + (2.32e10 - 2.32e10i)T - 1.71e21iT^{2} \)
47 \( 1 - 1.01e11T + 5.46e21T^{2} \)
53 \( 1 + (-6.00e10 + 6.00e10i)T - 2.60e22iT^{2} \)
59 \( 1 + (-3.60e10 + 3.60e10i)T - 1.04e23iT^{2} \)
61 \( 1 + (4.81e10 + 4.81e10i)T + 1.61e23iT^{2} \)
67 \( 1 + (4.87e11 + 4.87e11i)T + 5.48e23iT^{2} \)
71 \( 1 - 1.16e12iT - 1.16e24T^{2} \)
73 \( 1 + 1.30e12iT - 1.67e24T^{2} \)
79 \( 1 - 2.38e9T + 4.66e24T^{2} \)
83 \( 1 + (-9.08e11 - 9.08e11i)T + 8.87e24iT^{2} \)
89 \( 1 + 1.63e12iT - 2.19e25T^{2} \)
97 \( 1 - 7.23e11T + 6.73e25T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.07377117554071366377358139812, −14.28903942043302138512577403108, −12.66883218278825443052491912869, −11.40235107548677779349678134892, −10.40849122781844511371208249171, −7.46940815433793319953236506329, −6.52484936958568654982068394667, −4.21898239129473508250121085083, −3.10341199316113105980871520425, −0.58476980764682807600416829989, 2.17639556525441504484210358994, 4.33808075209461491779230309744, 5.29761839555497010057575950141, 7.38069579211827468900344295430, 8.962000010335304194730024462600, 11.50982047112338142854640365099, 12.11869745009569601941637560655, 13.52069139199611127697505178683, 15.31104179235323243224998059193, 15.90147217661223756266386271207

Graph of the $Z$-function along the critical line